Alternatively,if the channel is memoryless,and so the order of codeword bits is unimportant,a far easier option is to apply to the original Hto give a parity-check matrix 「00100110117 1100001101 H=0011100010 1101011000 1010100100 with the same properties as H but which shares the same codeword bit ordering as Had.在接 收端，我们使用H'对由G编出的码字进行译码。 All of this processing can be done off-line and just the matrices G and H'provided to the encoder and decoder respectively. The drawback of this approach is that,unlike H,the matrix G will most likely not be sparse and so the matrix multiplication in (4.4)have complexity in the order of noperations. 随着码长n的增加，编码复杂度会变得非常高。Richarson and Urbanke于2001年提出了 一种直接基于H矩阵的编码方法Richarson200L,T,它具有近似线性编码复杂度。另 种降低编码复杂度的方法是基于代数或几何方法构造LDPC码，这种结构化的校验矩阵 通常能够允许编码器采用简单的移位寄存器实现。下面我们首先介绍U的方法，具有 准循环H矩阵结构的QC-LDPC码编码方法见[19，Li-Lin2006]。 4.8.2.1 (Almost)Linear-Time Encoding of LDPC Codes Rather than finding a generator matrix for H,an LDPC code can be encoded using the matrix din 。uer ndinbok ng prior to encoding Firstly,we perform the following preprocessing steps.By row and column permutations, we bring H into the approximate lower triangular form as indicated in Fig.4.8.5,where the upper right comer can be identified as a lower triangular matrix.Because it is obtained only where T is a (m-g)x(m-g)lower triangular matrix with ones along the diagonal and hence is invertible.If H is full rank,the matrix B is of size (m-g)xg and A is of size (m-g)xk.The g rows of H left in C.D.and E are called the gap of this approximate representation.The smaller g the lower the encoding complexity for the LDPC code. 8 Alternatively, if the channel is memoryless, and so the order of codeword bits is unimportant, a far easier option is to apply π to the original H to give a parity-check matrix 0010011011 1100001101 0011100010 1101011000 1010100100 ⎡ ⎤ ⎢ ⎥ ′ = ⎣ ⎦ H with the same properties as H but which shares the same codeword bit ordering as Hstd. 在接 收端，我们使用H’对由G编出的码字进行译码。 All of this processing can be done off-line and just the matrices G and H′ provided to the encoder and decoder respectively. The drawback of this approach is that, unlike H, the matrix G will most likely not be sparse and so the matrix multiplication in (4.4) have complexity in the order of n 2 operations. 随着码长n的增加，编码复杂度会变得非常高。Richarson and Urbanke 于2001年提出了 一种直接基于H矩阵的编码方法[Richarson2001, IT]，它具有近似线性编码复杂度。另一 种降低编码复杂度的方法是基于代数或几何方法构造LDPC码，这种结构化的校验矩阵 通常能够允许编码器采用简单的移位寄存器实现。下面我们首先介绍R-U的方法，具有 准循环H矩阵结构的QC-LDPC码编码方法见[19, Li-Lin2006]。 4.8.2.1 (Almost) Linear-Time Encoding of LDPC Codes Rather than finding a generator matrix for H, an LDPC code can be encoded using the parity-check matrix directly by performing some preprocessing prior to encoding (transforming it into upper triangular form and using back substitution). Firstly, we perform the following preprocessing steps. By row and column permutations, we bring H into the approximate lower triangular form as indicated in Fig. 4.8.5, where the upper right comer can be identified as a lower triangular matrix. Because it is obtained only by permutations, the H matrix is still sparse. We denote the permutation decomposition as ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ ABT H CDE where T is a ( ) ( ) mg mg −× − lower triangular matrix with ones along the diagonal and hence is invertible. If H is full rank, the matrix B is of size ( ) mg g − × and A is of size ( ) mg k − × . The g rows of H left in C, D, and E are called the gap of this approximate representation. The smaller g the lower the encoding complexity for the LDPC code